ALIGNMENT BETWEEN FLATTENED PROTOSTELLAR INFALL ENVELOPES AND AMBIENT MAGNETIC FIELDS

The red and blue lines (vectors) in Figures 1–4 show the inferred magnetic field directions for each core in a manner similar to Figures 2–7 in Paper I. These inferred field directions are orthogonal to the measured directions of the 350 μm polarization. In the left panel, the vectors are plotted with lengths proportional to the percentage polarization. As in Paper I, we distinguish between high-flux and low-flux regions as a way to flag polarization measurements that may have large contamination from the parent cloud. Red vectors are used for sight-lines where the flux is greater than or equal to 25% of the peak flux, while blue vectors indicate sight-lines not meeting this threshold. The 25% flux cutoff matches the level used in Paper I.

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The left panel of each figure shows in gray scale a Spitzer 4.5 μm waveband image. This waveband is an excellent tracer of outflows because it contains a shocked H₂ emission line. For L483 and Serp-FIR1, images are from the final delivery of the Cores to Disks Legacy Science program¹⁴ while for L1157 and L1448-IRS2 images are from the Spitzer Heritage Archive.¹⁵ Overlaid on each Spitzer image are contours of the observed 350 μm emission, ranging from 20% to 90% of the peak flux in steps of 10%. The dashed circle and double-headed black vector indicate the infall radius and outflow position angle for each source, respectively (estimated from observations, see Table 3). The right panel plots the inferred magnetic field vectors superposed on Figure 8(c) from ALS03. Each vector is now shown with the same length to make the magnetic field morphology clearer. The vector locations are scaled to the model infall radius and the model is rotated by the pseudodisk position angle. It is important to note that there are no free parameters to help fit the magnetic field vectors to the ALS03 model. The infall radius and pseudodisk position angle are set by observations.

Also shown in each figure is the mean magnetic field direction that we computed for each core (white outlined double-headed arrow). In Paper I this mean was not computed, but we require it for the statistical analyses that we will carry out in Section 4.2 below. Because a polarization angle of 0° is the same as 180°, computing the mean polarization angle is non-trivial. We use the Equal Weight Stokes Mean, defined by Li et al. (2006). In brief, this method converts each angle to Stokes q = Q/I and u = U/I, computes the unweighted averages $\bar{q}$ and $\bar{u}$, and then converts $\bar{q}$ and $\bar{u}$ back to an angle. The uncertainty in mean angle is derived from propagation of errors. Only the red vectors in each core are used when computing the mean magnetic field angle.

ALS03 present several models corresponding to different values of magnetic field strength and rotation speed. Just as in Paper I, we show only one model, corresponding to Figure 8(c) of ALS03, chosen because it has intermediate values for both magnetization and rotation. However, note that all models in ALS03 (excluding the unmagnetized one) show a similar pinch in the field. Data from our survey cannot yet resolve the small differences between the various models in ALS03. Also, it is important to keep in mind that ALS03 display a slice through the center of their core. Averaging of the magnetic field along the line-of-sight will lessen the observed pinch in the field. Furthermore, large inclination angles of the pseudodisk symmetry axis will distort the observed pinch pattern. ALS03 show this distortion in their Figure 4 for inclinations of 30° and 90°. At an inclination of 30° the distortion is mild, but at 90° the magnetic field lines extend radially outward from the core center, with some twisting due to rotation.

L483 is located in the Aquila Rift at a distance of 200 ± 30 pc (Prato et al. 2008). The embedded protostar in L483 is IRAS 18148−0440. Based on its spectral energy distribution, it is usually classified as a Class 0 protostar (Fuller et al. 1995). However, Tafalla et al. (2000) argued that the outflow from the protostar shares some properties with those observed in more evolved sources and that IRAS 18148−0440 may thus be transitioning from Class 0 to Class I.

The inclination angle of the outflow in L483 is not well known. In the present paper we measure inclination from the plane of the sky; thus 0° inclination means the outflow is parallel to the sky plane and 90° means the outflow is pointing along the line-of-sight. Fuller et al. (1995) measured differences in 2.22 μm brightness between the eastern and western lobes of the outflow. By fitting this emission to a simple model, they estimated the inclination angle to be 40°. Based on their Figure 5 we estimate an uncertainty of ±10° on the inclination. This 40° inclination is the only quantitative value for the inclination angle in the literature, but it may be an overestimate. Hatchell et al. (1999) measured ¹²CO J = 4 → 3 emission from the outflow and found some overlap between the redshifted and blueshifted emission in each outflow lobe, which suggests that the L483 outflow has a smaller inclination angle. We estimate the position angle of the outflow to be 105° based on the shocked H₂ emission (Fuller et al. 1995).

There are no molecular line measurements of a pseudodisk in L483. This could be due to depletion and possibly also optical depth effects (Fuller & Wootten 2000; Park et al. 2000; Jørgensen 2004). Fuller & Wootten (2000) observed a narrow absorption lane at 3.4 μm with a width of ∼250 AU. From visual inspection of their Figure 4, we estimate that the diameter of this absorption lane must be ∼1000–2000 AU based on the apparent 3.4 μm extinction. The lane can also be seen at lower resolution in the Spitzer 4.5 μm emission in Figure 1. It has roughly the right size to be a pseudodisk. We measured its position angle to be 36°, which we adopt as the pseudodisk position angle.

There are two different estimates for the infall radius of L483. Myers et al. (1995) observed N₂H⁺ and C₃H₂ rotational emission lines at the source peak and nearby positions offset from the peak. They found that the centroid line velocities decreased (suggesting infall motion) as positions approached the protostar over a distance of 0.04 pc (40'' for a 200 pc distance). Tafalla et al. (2000) detected self-absorption in the H₂CO (2₁₂ − 1₁₁) emission toward L483, with a brighter blueshifted peak compared to the redshifted peak, indicative of infall motions. The radius over which they were able to measure a stronger blueshifted peak was 20''. Because the self-absorption profile requires specific physical conditions to occur, the lack of a stronger blueshifted peak is not necessarily evidence for lack of infall. Therefore, we adopt an infall radius of 40'' for this paper.

Figure 1 shows the inferred magnetic field in L483. The 20% contour of the 350 μm emission appears distorted due to artifacts at the edge of the map. The magnetic field is fairly ordered with a mean direction of 93° ± 6°. The mean field direction is offset by 12° from the direction of the outflow and offset by 33° from the symmetry axis of the pseudodisk. Considering the red and blue vectors together, there is a suggestion of a pinch in the magnetic field.

Dotson et al. (2010) reported polarimetric observations of L483 at 350 μm. They obtained several upper limits (p + 2σ_p < 1%) for sight lines near the flux peak and three polarization detections from the periphery (regions with I ≲ 20% of their peak flux). Our two vectors closest to the flux peak have polarization magnitudes of 1.0% and 0.8%, consistent with the upper limits of Dotson et al. (2010). Only one of the periphery vectors from Dotson et al. (2010) overlaps regions where we detect polarization; this occurs at (ΔR.A., Δdecl.) ∼ (+ 30'', −20''). We detect polarization at two locations near this position. Taken together, and considering the coarser angular resolution of Dotson et al. (2010), our vectors agree with their measurement in both polarization angle and polarization percentage.

The first distance estimate for L1157 was 440 pc, based on the distance to the NGC 7023 open cluster (Viotti 1969). More recently, Kun (1998) estimated distances of 200, 300, and 450 pc for different clouds in Cepheus. Because the Galactic latitude of L1157 is similar to that of the 200 pc and 300 pc clouds, some recent authors have assumed a distance of 250 pc for L1157 (Looney et al. 2007; Chiang et al. 2010; Tobin et al. 2010). Straizys et al. (1992) used an interstellar reddening–distance relationship to estimate a distance of 325 ± 13 pc for the L1147/L1158 subgroup (which includes L1157). The latter distance was used by the Spitzer Gould's Belt Legacy Science Program (Kirk et al. 2009) and is the distance we adopt here. L1157 contains the Class 0 protostar IRAS 20386+6751 (Gueth et al. 1997).

The outflow in L1157 has an inclination angle of 9° (Gueth et al. 1996). It has been mapped in shocked H₂ by Davis & Eisloeffel (1995) who found a position angle of 155°. L1157 has a flattened envelope roughly perpendicular to the outflow. It is very prominent in 8 μm absorption with a diameter of ∼2' (Looney et al. 2007). Chiang et al. (2010) observed N₂H⁺J = 1 → 0 and found it to spatially coincide with the absorption feature. The full width at half-maximum (FWHM) of the N₂H⁺ is ∼11'' × 18'', or ∼3600 × 5900 AU at the distance of L1157. The N₂H⁺ emission shows evidence of both infall and rotation (Chiang et al. 2010) and its size is not too different from the predicted pseudodisk size, so we adopt the position angle of the N₂H⁺ emission, which is 75°, as the pseudodisk position angle. Gueth et al. (1997) observed ¹³CO J = 1 → 0 and J = 2 → 1 transitions in L1157. Both spectra show a self-absorption profile with the blueshifted peak stronger than the redshifted peak, indicating infall (see their Figure 8). Using these two spectra, Gueth et al. (1997) estimated the path length toward the central protostar of the ¹³CO to be 8500 AU. We adopt this value as the infall radius for L1157.

We show our results for L1157 in Figure 2. Only one of our polarization detections has a corresponding flux value greater than or equal to 25% of the flux peak. This vector appears at the peak and has >3σ significance. It is offset by 13° with respect to the outflow axis and by 23° with respect to the pseudodisk symmetry axis. Hull et al. (2013) measured 1.3 mm polarization in L1157 on spatial scales much smaller than those studied in the present paper. Their mean magnetic field is consistent with our measurement, and will be discussed in more detail in Section 5.

The L1448 complex is located in the Perseus cloud. L1448-IRS2 lies approximately 3' west of L1448-mm/L1448C and L1448-IRS3 and 3' east of L1448-IRS1. In addition, roughly 50'' east of IRS2 is a candidate first hydrostatic core labeled L1448-IRS2E (Chen et al. 2010). Hirota et al. (2011) obtained a distance of 232 ± 18 pc toward H₂O masers in L1448C. Because of L1448C's proximity, we adopt this distance for L1448-IRS2. L1448-IRS2 contains the protostar IRAS 03222+3034. Based on its spectral energy distribution it is a Class 0 source (O'Linger et al. 1999).

The position angle for the outflow is 138° as derived from shocked H₂ emission (Davis et al. 2008). Less certain is the inclination angle of the outflow. Tobin et al. (2007) used radiative transfer codes (Whitney et al. 2003a, 2003b) to model continuum emission data covering the wavelength range 2.2 μm to 2.7 mm. The best fit inclination angle is $33^{+8}_{-6}$ degrees. A possible pseudodisk has been observed in 1.3 mm continuum emission by both Kwon et al. (2009) and Chen et al. (2010) with 5'' and 3'' resolution, respectively. In both maps, the core appears extended with a long axis oriented at position angle 45°. We measure the length of the long axis as ∼14'' from the data of Kwon et al. (2009) and ∼5'' from the data of Chen et al. (2010). Thus the spatial size is ∼1000–3000 AU for a distance of 232 pc. There are no direct measurements of the infall radius for this source. However, the best-fit model from Tobin et al. (2007) has an envelope radius of 8000 AU. Because the Whitney et al. (2003a, 2003b) models use an infalling envelope, we adopt the envelope radius as the infall radius for L1448-IRS2.

In Figure 3 we plot our results for L1448-IRS2. We pick up some emission from L1448-IRS2E at the edges of our map. This emission is the primary reason for the distorted 20% contour. Taken together, the red and blue vectors are very ordered with a uniform direction. There is no evidence of a pinch in the magnetic field. Note that the two easternmost vectors arise from the wings of L1448-IRS2E. To avoid any possible bias, we exclude these vectors and use only the remaining four red vectors in computing the mean field direction. The mean field direction is 147° ± 5° which is offset by 9° from the outflow axis and by 12° from the pseudodisk symmetry axis. Hull et al. (2013) measured 1.3 mm polarization in L1448-IRS2 on spatial scales much smaller than those studied in the present paper. Their mean magnetic field is consistent with our measurement, and will be discussed in more detail in Section 5.

Serp-FIR1 (also known as Serp-SMM1) is located in the Serpens core. Most distance estimates for Serpens are in the range 200–400 pc (Eiroa et al. 2008). In the present paper we adopt a distance of 415 ± 5 pc based on the work of Dzib et al. (2010), who measured the trigonometric parallax for the binary YSO EC95, located in the Serpens cloud core. Serp-FIR1 is the brightest source in the Serpens core at submillimeter wavelengths and contains the Class 0 protostar IRAS 18273+0113 (Hurt & Barsony 1996).

The Serp-FIR1 outflow has been observed in 6 cm continuum with the Very Large Array. The continuum is resolved into three peaks, one centered on the source, the other two offset by ∼6'' on either side (Rodriguez et al. 1989). The three peaks form a straight line with a position angle of 130°. The inclination angle of the outflow is not well determined. Enoch et al. (2009) performed radiative transfer modeling of continuum emission in multiple wavebands from 3.6 μm to 1 mm. Their best-fit inclination angle for the outflow is 75°, although angles in the range 65°–80° also produce reasonable fits. We adopt an inclination angle of 725 ± 75. No candidate pseudodisk has been observed for Serp-FIR1. High-resolution interferometric observations show a very round or unresolved core at 3 mm (Williams & Myers 2000; Hogerheijde et al. 1999; Enoch et al. 2009) and also at 1 mm (Hogerheijde et al. 1999; Enoch et al. 2009). If the pseudodisk axis is close to the outflow axis (see Section 4.1), then this could be a projection effect; a flattened envelope would appear round if viewed face-on. There are no measurements of an infall radius for Serp-FIR1. Because the model from Enoch et al. (2009) is of a rotating, collapsing sphere, we adopt their best-fit envelope radius of 5000 AU.

Our results for Serp-FIR1 are shown in Figure 4. Despite the high inclination angle for the Serp-FIR1 outflow, for completeness we plot the vectors on the ALS03 model in Figure 4. The position angle of the pseudodisk plane is taken to be perpendicular to the outflow axis since no pseudodisk is detected (see discussion in Section 4.1). The magnetic field is well ordered. Its mean direction is 62°  ±  5°, nearly perpendicular to the outflow axis, and thus also to the assumed pseudodisk symmetry axis. This may indicate that magnetic fields do not regulate star formation in this core. Alternatively, it may be a projection effect caused by the outflow pointing nearly parallel to the line-of-sight. To understand this, note that the perceived angle between two vectors is heavily dependent on viewing angle. This can be readily seen by considering the situation depicted in Figure 5. An observer along the z-axis will measure the projected separation angle between the two vectors to be 90°, even though the true angle between them is much smaller. We conclude that the high inclination of the Serp-FIR1 outflow makes this source a poor test of the basic predictions of magnetically regulated core-collapse (see Section 1). We will further explore the effects of viewing angle on projected separation in Section 4.2.

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Polarization toward Serp-FIR1 has been detected at 850 μm by Matthews et al. (2009). In general, their polarization measurements trace the magnetic field in the cloud rather than in the Serp-FIR1 core. All but one of their vectors correspond to regions where the measured 350 μm flux is less than 20% of the peak flux. This one vector implies a magnetic field position angle of 633, almost identical to our mean field direction. Hull et al. (2013) measured 1.3 mm polarization in Serp-FIR1 on spatial scales much smaller than those studied in the present paper. Their mean magnetic field direction differs by nearly 90° from our own. This will be discussed in more detail in Section 5.

We find a strong correlation between the projected outflow axis and the projected pseudodisk symmetry axis for all cores with a measured pseudodisk (i.e., all except Serp-FIR1). Using data from Table 3 of the present paper and Table 2 and Section 3.1 of Paper I, we find that the differences in angle between these two axes are: 21° (L483), 10° (L1157), 3° (L1448-IRS2), 0° (L1527), 17° (IC348-SMM2), and 20° (B335). The mean of these six values is 12°. This tight correlation gives us confidence in using the outflow axis as a proxy for the pseudodisk symmetry axis in Serp-FIR1, as we did in Section 3.5 above. In Section 4.2 we will test for a correlation between the pseudodisk and magnetic field directions. For this purpose we will need the inclination of the pseudodisk axis. Because this is unknown, we will exploit the tight correlation between pseudodisk and outflow axes by using the inclination of a given source's outflow as a proxy for the inclination angle of that source's pseudodisk axis. Furthermore, for Serp-FIR1 (only) we will again use the position angle of the outflow as a proxy for the position angle of the pseudodisk axis, while for the remaining six sources we will use our measured pseudodisk axis position angles.

We define ϕ to be the projected plane of sky separation angle between the mean magnetic field and the pseudodisk symmetry axis. The value of ϕ for each core is listed in Table 4. Since these values range from 9° to 68°, it appears that any correlation is much less evident than the correlation we found between outflow and pseudodisk axis. However, six of the seven ϕ values are less than 45°, and the remaining value corresponds to Serp-FIR1, which, as we discussed in Section 3.5, has high outflow inclination making this source a poor test for intrinsic three-dimensional (3D) alignment. In the remainder of this section, we will explore quantitatively a possible correlation between the 3D core magnetic field angle and the 3D pseudodisk symmetry axis for our sample of seven sources. We define α to be the true 3D separation angle between magnetic field and pseudodisk symmetry axis and i to be the inclination angle of the pseudodisk symmetry axis. (Recall that we will use outflow inclination as a proxy for pseudodisk inclination, as discussed in Section 4.1.) In our analysis, we assume that each source has the same α, and we use our data to obtain a best estimate for this parameter.

Consider two vectors: p, representing the pseudodisk symmetry axis, and b, representing the mean magnetic field direction. Figure 6 shows the coordinate system and nomenclature used. Vectors p and b can be written in xyz components as

$$\begin{eqnarray} \boldsymbol p &=& p(\sin \theta _1,\:\: 0,\:\: \cos \theta _1) \end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray} \boldsymbol b &=& b(\sin \theta _2\cos \phi,\:\: \sin \theta _2 \sin \phi,\:\: \cos \theta _2). \end{eqnarray} \tag{ 2 }$$

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Now consider a new coordinate system that is rotated about the y-axis by θ₁ degrees such that the new z-axis, which we call z', is aligned with p. Note that under this rotation y' = y. In the new x'y'z' coordinate system b is written as

$$\begin{eqnarray} \boldsymbol b & = & b(\cos \theta _1 \sin \theta _2 \cos \phi \nonumber\\ && -\sin \theta _1 \cos \theta _2,\:\: \sin \theta _2\sin \phi, \:\: \sin \theta _1 \sin \theta _2 \cos \phi \nonumber \\ && +\cos \theta _1 \cos \theta _2) \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} & \equiv & b (\sin \alpha \cos \phi ^{\prime },\:\: \sin \alpha \sin \phi ^{\prime },\:\: \cos \alpha), \end{eqnarray} \tag{ 4 }$$

where in the primed coordinate system α takes the place of θ₂ and ϕ' takes the place of ϕ. If one holds α fixed while rotating b about p, then ϕ' varies from 0° to 360°. We can set the three components of b equal to each other:

$$\begin{eqnarray} \sin \alpha \cos \phi ^{\prime } & = & \cos \theta _1 \sin \theta _2 \cos \phi - \sin \theta _1 \cos \theta _2 \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} \sin \alpha \sin \phi ^{\prime } & = & \sin \theta _2 \sin \phi \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \cos \alpha & = & \sin \theta _1 \sin \theta _2 \cos \phi + \cos \theta _1 \cos \theta _2. \end{eqnarray} \tag{ 7 }$$

Using these three equations we can then solve for the projected separation, ϕ:

$$\begin{equation} \tan \phi = \frac{\tan \alpha \sin \phi ^{\prime }}{\sin \theta _1 + \cos \theta _1 \tan \alpha \cos \phi ^{\prime }}. \end{equation} \tag{ 8 }$$

Therefore, by assuming an inclination angle i, where θ₁ = 90 − i, and assuming a separation angle α, we can compute ϕ as a function of ϕ'. As ϕ' varies from 0° to 360°, ϕ will take on a range of values. The probability density function (PDF) for ϕ can then be derived by making the reasonable assumption that all values of ϕ' are equally likely.

We computed the PDFs for three values of α and 901 equally spaced values of i (0°–90° in steps of 01). In Figure 7 we show these results as three plots, one for each value of α. The PDFs are shown in gray scale where dark means most likely value of ϕ for each i. The black curve shows $\bar{\phi }(i,\alpha)$, the mean value of ϕ for each i and α (averaged over all ϕ') and the lower (upper) gray curve marks the value of ϕ lying above 10% (90%) of the integrated probability for each i. The data for our seven cores, taken from Table 4, are superposed on the probability density maps. Recall that for each core the outflow inclination has been used as a proxy for the unknown pseudodisk inclination i (see Section 4.1).

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From an examination of Figure 7 it is clear that some values of α fit the data better than do others. For α = 20° some points appear in a region forbidden by the models and are far away from the average curve. The opposite extreme happens for α = 50°, where two of the points lie below the 10% curve. Qualitatively, α = 35° appears more likely since all points lie near the average curve. Two points lie in the forbidden region, but they are very close to the border, where the PDF is largest.

A rigorous analysis aimed at finding the best-fit α by taking into account the full density distribution as well as the uncertainties in both projected separation ϕ and inclination i is beyond the scope of this paper. Instead, we will crudely estimate the best-fit α by minimizing the chi-squared difference between our data and $\bar{\phi }(i,\alpha)$. We created a grid of models with α ranging from 0° to 90° in steps of 01. For each model we computed χ², defined as

$$\begin{equation} \chi ^2 = \sum _{j=1}^7 \frac{[\phi _j - \bar{\phi }(i,\alpha)]^2}{\sigma _{\phi _j}^2}, \end{equation} \tag{ 9 }$$

where ϕ_j is the projected separation for an individual source and $\sigma _{\phi _j}$ is the uncertainty in the projected separation. For this simple analysis we set all the $\sigma _{\phi _j}$ to unity when computing χ², thereby giving each point equal weight. Thus, our simple approach amounts to a least-squares fit to the average curve $\bar{\phi }(i,\alpha)$. The model with the smallest χ² has α = 350, which is our initial crude estimate for the best-fit value of α.

Our analysis method may lead to a bias in our best-fit value for α. To see this, note that it favors models in which the data points lie close to the average curve, whereas in reality we expect the data points for a given i to follow a distribution given by the PDF. Our crude approach could thus penalize high values for α, for which the data are expected to have a relatively larger spread from the model average curve (see Figure 7). To explore the magnitude of this possible bias, we carried out Monte Carlo simulations as follows. We adopted an assumed value α_true, then using the measured inclinations i for our seven sources, we computed from Equation (8) seven values of the projected separation ϕ by giving each source a random ϕ'. We then fit these simulated data to find α_fit. We repeated this 10,000 times each for α_true in the range 20° to 50° in steps of 1°. We then computed the mean α_fit for each input α_true and fit the data to a straight line to obtain the following relation:

$$\begin{equation} \alpha _{{\rm true}} = 0.91\alpha _{{\rm fit}} + 4\buildrel{\circ}\over{.}24. \end{equation} \tag{ 10 }$$

The mean absolute difference between α_true computed via Equation (10) and the value input to the Monte Carlo simulations is only 026. Starting with our initial best-fit α of 350, we used Equation (10) to find a bias-corrected best-fit α of 361.

We are now in a position to compute the probability of obtaining a bias-corrected best-fit α at or below 361 by pure chance. We consider the case where b points in a random direction in 3D space compared to p (see Figure 6). Note that this is not equivalent to choosing uniformly distributed random values for α and ϕ' because such a distribution will sample the unit sphere non-uniformly (the differential unit of surface area is sin α dα dϕ'). We found that when α and ϕ' are chosen such that the unit sphere is sampled uniformly, then the resulting distribution of ϕ (computed from Equation (8)) is also uniform, regardless of θ₁. Therefore, generating random directions of b reduces to merely generating random values of ϕ directly.

We ran 10,000 random b Monte Carlo simulations. In each simulation, we held the inclination angles of each of our seven cores fixed at the values shown in Table 4 and Figure 7, and for each core we chose a random value of the projected separation ϕ (0° to 90°, uniform distribution). Then we computed the best-fit α for each simulation. Only 4.21% of the models yielded a bias-corrected best-fit α at or below 361. Alternatively, 95.79% of models with random projected separation angles led to a best-fit α greater than 361. Therefore, our analysis indicates that we have detected a positive correlation between the core magnetic field direction and the pseudodisk symmetry axis with ∼96% confidence. Additional sources of error are considered in Section 4.5 below, leading to some modifications to the above conclusions regarding α and our confidence level.

In the previous subsection we utilized the mean magnetic field direction for each of our seven cores, as computed using 2σ polarization measurements from Table 2 of the present paper and Table 1 of Paper I. However, our Stokes core maps hold more information than what is contained in just the points with 2σ detections. We can make use of this information by combining the maps for many cores into a single source-averaged magnetic field map. Such a map will naturally have an enhanced signal-to-noise ratio. Our goal is to compare this combined map to Figure 8(c) of ALS03, which is the model shown in Figures 1–4. As discussed in Section 3.5, the high inclination angle of Serp-FIR1 makes this core a poor test of magnetically regulated core-collapse, and also makes it unsuitable for overlaying on Figure 8(c) of ALS03. For these reasons, it is excluded from the following analysis.

We start with the combined maps for Stokes I, Q, and U and their associated errors for each core (see Section 2). We exclude sky positions where the flux is less than 25% of the peak, just as was done in Paper I and in earlier sections of the present paper. Next, we rotate the pixel positions in each core so that the pseudodisk lies horizontal and the blueshifted lobe of the molecular outflow lies in the top half of the rotated figure. Note that when rotating Q and U by an angle θ, the values become intermixed by a rotation matrix having angle 2θ:

$$\begin{equation} \left[\begin{array}{@{} c @{}}Q^{\prime }\\ U^{\prime } \end{array} \right] = \left[\begin{array}{@{} cc @{}}\cos 2\theta & {-}\sin 2\theta \\ \sin 2\theta & \phantom{-}\cos 2\theta \end{array} \right] \left[\begin{array}{@{} c @{}}Q \\ U \end{array} \right]. \end{equation} \tag{ 11 }$$

We next compute q = Q/I and u = U/I and their associated errors for each core.

Just as for Figures 1–4, our magnetic field maps must be scaled to account for distance and infall radius before they can be compared with the ALS03 model. Thus, we convert the rotated pixel offsets for each core onto the same scale by dividing the pixel scale in arcseconds (95) by the core's infall radius measured in arcseconds. The computed pixel scale for each core is given in Table 4. With the combined data from all six cores we have many independent polarization measurements all ready for superposition onto the ALS03 model. Next these measurements need to be combined and sampled on a uniform grid to increase the signal-to-noise.

To combine all these polarization measurements into one polarization map we overlay a grid on the data and at each grid point compute the weighted averages $\bar{q}$ and $\bar{u}$ as well as the associated errors $\sigma _{\bar{q}}$ and $\sigma _{\bar{u}}$. We use a Gaussian kernel with weighting based on distance from the grid point and the errors in the individual measurements. The kernel FWHM is set equal to the grid spacing and we use a cutoff radius also equal to the grid spacing. We have freedom in choosing the grid spacing, so we opt to use both extremes from Table 4, yielding two different source-averaged maps. L483 has the smallest grid spacing (highest resolution) while IC348-SMM2 has the largest spacing (lowest resolution). B335, L1448-IRS2, and L1527 have spacings similar to that of L483 while L1157's spacing is close to that of IC348-SMM2.

After creating regularly sampled maps of $\bar{q}$, $\bar{u}$, $\sigma _{\bar{q}}$, and $\sigma _{\bar{u}}$ we compute the polarization and magnetic field direction using the same techniques, cutoffs, and debiasing as for the analysis described in Section 2. We obtain 35 vectors for the high resolution map and 26 for the low resolution map. These magnetic field maps are shown in Figure 8. The gray circles at the bottom right of each panel show the Gaussian kernel used to create each map. We also compute and plot the mean magnetic field direction (Equal Weight Stokes Mean; see Section 3.1). The mean field angle is 166° for the data in Figure 8(a) and 169° for Figure 8(b).

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The combining of our sources to produce a source-averaged magnetic field map for a Class 0 protostar gives us a better sampling of the characteristic magnetic field structure than was possible for any individual source. The appearance of Figure 8 is consistent with our result from the previous subsection; the mean magnetic field direction is nearly parallel to the pseudodisk symmetry axis (within 15° for both the low- and high-resolution maps). Furthermore, we see hints of a pinch in the field, as predicted by magnetically regulated core-collapse models. The pinch appears to be stronger than predicted on the right side of each map and weaker than predicted on the left side. Nevertheless, all four quadrants of each map show a tendency for the field to be drawn inward as we approach the pseudodisk plane, in qualitative agreement with model predictions.

Because the outflow axis is strongly correlated with the pseudodisk symmetry axis (Section 4.1) and the pseudodisk symmetry axis is preferentially aligned with the magnetic field (Section 4.2), we anticipate a positive correlation between outflow axis and magnetic field. We can address this quantitatively by repeating the analyses of Sections 4.2 and 4.3 using the outflow axis instead of the pseudodisk symmetry axis. When computing the projected separation ϕ using the outflow axis, we find the bias-corrected best-fit value of α_o to be 344 (here denoted α_o to distinguish it from α computed using the pseudodisk). From Monte Carlo simulations we estimate the probability of obtaining such a value by chance to be 2.87%, corresponding to a confidence level of ∼97%. These results are nearly the same as the values obtained using the pseudodisk axis, which were α = 361 and ∼96% confidence. We also stacked and combined the cores, as in Section 4.3, but this time rotated each core so that the outflow axis points straight up and down with the blue lobe of the molecular outflow still in the top half of the rotated figure. The mean magnetic field angle is 160° at high resolution and 165° at low resolution. These angles are similar to the 166° and 169° values obtained when setting the observed pseudodisk major axis to be exactly horizontal. Furthermore, the magnetic field maps are nearly identical to those shown in Figure 8. We conclude that we have found evidence of a correlation between magnetic field direction and outflow axis, with a similar degree of confidence as for the pseudodisk-magnetic field correlation discussed earlier.

One possible source of error is the slight difference in selection of vectors between Paper I and the present paper. Paper I considered vectors having p/σ_p ⩾ 2 before debiasing to be polarization detections while in the present paper we require vectors to pass that threshold after debiasing. Since debiasing lowers p/σ_p, when we computed the mean magnetic field angles for Paper I cores we used some vectors that would not have been selected under our new criterion. If we require p/σ_p ⩾ 2 after debiasing for the cores in Paper I, we lose five vectors (one in L1527, three in IC348-SMM2, and one in B335). Without these vectors, the mean magnetic field angle changes from 49° to 47° in L1527 and from 137° to 147° in IC348-SMM2. There is only one high-flux vector in B335, so applying the stricter criterion results in the loss of this source for analysis, leaving only six sources for fitting α. The bias-corrected best-fit α decreases to 315 (∼97% confidence) and the bias-corrected best-fit α_o decreases to 262 (∼98% confidence). Thus, if we apply the stricter selection criterion to the data from Paper I, the statistical significance of the correlation between mean magnetic field direction and pseudodisk symmetry axis increases slightly. The same is true for the correlation between mean field direction and outflow axis.

In Section 2 we discussed the improved error analysis used in the present paper. Is it possible that this improved error analysis, if applied to the data in Paper I, would significantly alter those results? To test this, we reprocessed the data for L1527 using the new method. There are seven polarization detections in common between the old and new methods. The median difference in angle for these seven polarization vector pairs is 5°, which is less than the median uncertainty of 10°. Furthermore, the difference in mean magnetic field angle is less than one degree. Therefore, it does not appear that the change in error analysis method between Paper I and the present paper significantly impacts our results.

Observational uncertainties in inclination angle and projected separation are another source of error. Small variations in these quantities may lead to large changes to our best-fit α. To test the robustness of our fits, we again ran Monte Carlo simulations. In each simulation, each source's inclination and projected separation angle were varied randomly using a Gaussian weighting with standard deviation equal to the corresponding errors listed in Table 4. The resulting mean and standard deviation of the bias-corrected best-fit values of α and α_o are 354 ± 39 and 346 ± 45, respectively. Therefore, our best-fit α and α_o are robust to within ∼4°. Uncertainties in inclination angle may also impact our estimated confidences. We re-ran the Monte Carlo simulations described in the last two paragraphs of Section 4.2, this time allowing the inclination angles to vary with a random Gaussian weighting. The probability of obtaining a bias-corrected best-fit α of less than or equal to 361 by pure chance increases to 5.25%, and the probability for α_o increases to 3.92%. Therefore, we downgrade our earlier confidence levels for the pseudodisk-magnetic field and outflow-magnetic field correlations to 95% and 96%, respectively.

Poor sampling of a pinched magnetic field can lead to error in computing the mean field. For example, if in Figure 8 the magnetic field were to be only measured in the upper left and bottom right quadrants, then the computed mean field direction would be biased counterclockwise. Such an error may have occurred for L1527 and B335 (shown in Figures 5 and 7 of Paper I, respectively). If the field in these cores is accurately described by the ALS03 model aligned with the pseudodisk symmetry axis, then the computed mean magnetic field direction in both cores is rotated away from the symmetry axis of the magnetic field by a large angle. This would lead to too-large projected separation values, ϕ, for these cores. Fitting artificially high values of ϕ at low inclination would lead to an overestimate of α. Because of this possible source of error, our best-fit values for α and α_o should be treated as rough upper limits rather than best estimates. Folding in the uncertainties discussed in the previous paragraph, we estimate an upper limit on α of 354 + 39 or ∼39°. Similarly, our estimated upper limit on α_o becomes 346 + 45 or ∼39°.